# Find an upper bound for an objective function

My objective function is $$\log_2(1+{x^2y^2})$$ and I found two upper bounds for $$x^2$$ and $$y^2$$.

For example, assumed that we have the following upper bounds: $$x^2\leq\text{constant}_1^2$$ and $$y^2\leq\text{constant}_2^2$$.

So, my question is: could we say that $$\log_2(1+{x^2y^2})\leq \log_2(1+{\text{constant}_1^2\cdot\text{constant}_2^2})$$? If so, how tight is this bound?

• I would start with using $x^2y^2$ as objective. Apr 11 at 19:37
• Actually, the main objective function is $\sum_{i}\log_2(1+x_i^2y_i^2)$ Apr 12 at 9:44

Yes, because $$\log$$ is monotonic, it preserves inequalities. The tightness depends on your other constraints.